PHY401: Nuclear and Particle Physics Lecture 13, Monday, September 21, 2020 Dr. Anosh Joseph IISER Mohali Radioactivity PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Radioactivity A nuclide with a low binding energy(“parent nucleus”) would prefer to decay... ... into one with a higher binding energy(“daughter nucleus”). Since it is energetically favorable. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Radioactivity This decay process can happen by 1. α-decay: the nuclide releases an an α-particle (helium nucleus). 2. β- decay: an electron and an anti-neutrino are emitted. 3. β+ decay: a positron and a neutrino are emitted. 4. γ-decay: a nucleus in a metastable excited state (“isomer”) decays directly or indirectly to its ground state emitting one or more high energy photons (γ-rays). PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Radioactivity In the last lecture we saw the decay equation -λt N(t) = N0e . (1) N0: number of parent nuclei at time t = 0. τ: mean lifetime. That is, time taken for the number of parent nuclei to fall to1 =e of its initial value. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Radioactivity From Eq. (1) we see that 1 τ = . (2) λ The “half-life” of a radioactive nucleus, τ 1 is the time 2 taken for the number of parent nuclei to fall to one-half of its initial value. From Eq. (1) we see that ln 2 τ 1 = = τ ln 2. (3) 2 λ PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Carbon Dating Life on earth is carbon based. 14 Living organisms absorb the carbon isotope 6 C. This isotope is created in the atmosphere by cosmic ray activity. 14 The production of 6 C from cosmic ray bombardment roughly cancels the rate at which that isotope decays. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Carbon Dating 14 Thus the global concentration of 6 C remains almost constant. The totality of atmosphere, biosphere, and oceans is known as the carbon exchange reservoir. The concentration ratio between C-14 and C-12 is approximately the same throughout the reservoir. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Carbon Dating A sample of carbon taken from a living organism will have a C-14 concentration of one part in 1.3 × 1012. The living organism is being continually rejuvenated, by exchanging carbon with the environment - by photosynthesis, or by eating plants or by eating other animals. On the other hand, a sample of carbon from a dead object cannot exchange its carbon with the environment and therefore cannot rejuvenate its concentration of C-14. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Carbon Dating 14 C-14 decays radioactively into 7 N (nitrogen), via β-decay with a half-life of 5730 years. Thus, by measuring the concentration of the isotope C-14 in a fossil sample,... ... using techniques of mass spectroscopy, the age of the fossil can be determined. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Multi-modal Decays There are situations in which a radioactive nucleus can sometimes decay into more than one channel, each of which has its own decay constant. When a nuclide can decay by more than one pathway, it is said to exhibit multi-modal decay. In this situation, the total decay constant is given by the sum of decay constant λ1 along branch 1 and decay constant λ2 along branch 2. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Multi-modal Decays Parent nuclei decays according to dN(t) = -λ N(t)- λ N(t). (4) dt 1 2 Solution is -(λ1+λ2)t N(t) = N0e . (5) PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Multi-modal Decays Let us consider an example. 212 Bismuth-212 ( 83 Bi) can decay as 212 208 83 Bi ! 81 Tl + α, (6) 212 212 - 83 Bi ! 84 Po + e + ν¯, (7) with a total mean lifetime of 536 seconds. (Tl: Tellurium, Po: Polonium.) 208 212 Ratio of 81 Tl to 84 Po from these decays is found to be 9:16. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Multi-modal Decays What are the decay constants λ1 and λ2, for each of these decay modes? We have 1 λ + λ = = 1.86 × 10-3s-1. (8) 1 2 536 Ratio of the number of decay products is equal to the ratio of the decay constants. λ 9 1 = . (9) λ2 16 PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Multi-modal Decays This gives -4 -1 λ1 = 6.8 × 10 s , (10) -3 -1 λ2 = 11.8 × 10 s . (11) PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains There are cases in which a parent nucleus decays, with decay constant λ1 into a daughter nucleus,... ... which is itself radioactive, and decays (either into a stable nuclide or into another radioactive nuclide) with decay constant λ2. An example of this is 210 β 210 α 206 83 Bi −! 84 Po −! 82 Pb (12) PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains Mean lifetime for the first stage of decay is 7.2 days. Mean lifetime for the second stage is 200 days. Suppose at time t we have N1(t) nuclei of the parent nuclide and N2(t) nuclei of the daughter nuclide. Then for N1(t) we have dN (t) 1 = -λ N (t). (13) dt 1 1 PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains This gives -λ1t N1(t) = N1(0)e . (14) For N2 there is a production mechanism,... ... which contributes a rate of increase of N2 equal to the rate of decrease of N1. In addition, there is a contribution to the rate of decrease of N2 from its decay process. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains Thus we have dN (t) 2 = λ N (t)- λ N (t). (15) dt 1 1 2 2 That is, dN2(t) = λ N (0)e-λ1t - λ N (t). (16) dt 1 1 2 2 PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains This is an inhomogeneous differential equation whose solution with N2(0) = 0 is given by λ 1 -λ1t -λ2t N2(t) = N1(0) e - e . (17) (λ2 - λ1) This equation is known as the Bateman equation. (It is in fact a simple form of the set of Bateman equations. Here this equation is for a chain of three isotopes.) PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains Initially, as the parent decays, the quantity of the daughter nuclide grows faster than it decays. However, after some time, the available quantity of the parent nuclide is depleted... ... so the production rate decreases. This in turn allow the decay rate of the daughter nuclide to dominate so that the quantity of the daughter nuclide also decreases. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains Some heavy nuclides have a very long decay chain, decaying at each stage to another unstable nuclide before eventually reaching a stable nulcide. An example of this is U-238. It decays in no fewer than 14 stages - eight by α-decay and six by β-decay before reaching a stable isotope of Pb. The lifetimes for the individual stages vary from around 10-4 s to 109 years. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains Note: Since uranium compounds are common in granite, uranium and its radioactive daughters are a part of the stone walls of buildings. They therefore contribute to the environmental radiation background. This is particularly true of the inert gas Rn-222 (radon), which escapes from the walls and is inhaled into the lungs. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains The α-decay of Rn-222 is responsible for about 40% of the average natural human radiation exposure. Figure below: Illustration of the U-238 decay chain in the N-Z plane. The half-life of each of the nuclides is given together with its decay mode. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Decay Chains PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Induced Radioactivity It is possible to convert a nuclide which is not radioactive into a radioactive one by bombarding it with neutrons or other particles. The stable nuclide (sometimes) absorbs the projectile in order to become an unstable, radioactive nucleus. 23 For example, bombarding 11Na (sodium) with 24 neutrons can convert the nuclide to 11Na. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Induced Radioactivity 24 It is radioactive and it decays to 12Mg (magnesium) through β-decay. Let us assume that the rate at which the radioactive nuclide (with decay constant λ) is being generated is R. Then the number of such nuclei is given by dN(t) = R - λN. (18) dt PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Induced Radioactivity If we take the number of these nuclei as zero at time t = 0 (that is, we start the bombardment at t = 0) then the above equation has the solution R N(t) = (1 - e-λt ). (19) λ At large t we have R = λN. (20) That is, we achieve the equilibrium state in which the production rate R is equal to the decay rate λN. PHY401: Nuclear and Particle Physics Dr.